See Section 14 of the Companies Act 2006:
If the registrar is satisfied that the requirements of this Act as to
registration are complied with, he shall register the documents
delivered to him.
The requirements are set out here. They are based on various documents being "delivered to" companies house, and various statements that those documents must contain. By way of Section 14 above, if documents containing those statements are delivered to Companies House, then it must unconditionally register the company.
We could form a counter-argument. One of the statements that must be on the application, under Section 9(5) is:
a statement of the intended address of the company's registered
office, which must be an appropriate address within the meaning given
by section 86(2)
The part in italics was only recently added to the Act, so it's possible that Companies House might take a different position going forward. The article you linked to is from before the change.
Section 86(2) defines an "appropriate address" as follows:
An address is an “appropriate address” if, in the ordinary course of
events — (a) a document addressed to the company, and delivered there
by hand or by post, would be expected to come to the attention of a
person acting on behalf of the company [...]
Taking all of the above, if the application uses someone else's address then it won't be an appropriate address, the requirements for registration will not have been met, and Companies House could refuse registration. It's not clear to me however that this gives Companies House the right to verify the accuracy of the address.
Regardless of whether they have the right to investigate, they certainly don't have the obligation. That's because of Section 13:
(1)The statement of compliance required to be delivered to the
registrar is a statement that the requirements of this Act as to
registration have been complied with.
(2)The registrar may accept the statement of compliance as sufficient
evidence of compliance.
The UK has a reputation for being a "business friendly" jurisdiction where there is relatively little red tape or expense involved in setting up a company. A big part of that is that the process of registration is straightforward, both for the new company and Companies House; it's essentially automated in most cases. This in turn enables Companies House to keep their fees extremely low. Prior to 1 May 2024, it only cost £12 to register a company. Even now it only costs £50. Those kinds of costs might be very difficult to maintain if Companies House had to take manual steps to verify information on each application it receives. I suspect this is the (non-legal) reason why they haven't implemented such a policy so far.